PUMaC 2020 · 加试 · 第 6 题
PUMaC 2020 — Power Round — Problem 6
题目详情
- When a problem asks you to “find”, “find with proof,” “show,” “prove,” “demon- strate,” or “ascertain” a result, a formal proof is expected, in which you justify each step you take, either by using a method from earlier or by proving that everything you do is correct. When a problem instead uses the word “explain,” an informal expla- nation suffices. When a problem instead uses the word “sketch” or “draw” a clearly marked diagram is expected.
解析
Problem 6.1. Let P be a polygon which has a billiard trajectory not passing through any vertices of P . Then, prove that for all small enough ε there is an ant-path on the surface of Q which passes through one of the bases. Conversely, prove that if for all small enough ε ε there is an ant-path on the surface of Q which passes through one of the bases, then there ε is a billiard trajectory on P not passing through any vertices of P . Team Number: PUMaC 2019 Power Round Cover Sheet Remember that this sheet comes first in your stapled solutions. You should submit solutions for the problems in increasing order. Write on one side of the page only. The start of a solution to a problem should start on a new page. Please mark which questions for which you submitted a solution to help us keep track of your solutions. Problem Number Points Solution written 1.1 10 y 1.2 20 y 1.3 20 y 1.4 15 y 2.1 20 y 2.2 20 y 2.3 20 y 2.4 20 Daniel 2.5 30 y 3.1 10 y 3.2 20 y 3.3 5 y 3.4 35 y 3.5 25 Aleksa 3.6 30 y 4.1 10 y 4.2 20 y 4.3 30 y 4.4 20 y 4.5 60 Aleksa 4.6 50 Aleksa 4.7 50 Aleksa 5.1 10 y 5.2 20 y 5.3 10 y 5.4 40 Aleksa 5.5 20 y 5.6 30 y 5.7 30 y 5.8 30 Daniel 5.9 60 Daniel 5.10 20 Daniel